Is there an easy way to tell if I am SSH'd in somewhere without typing 'exit'?

Like, if I am currently sitting at computer A and I SSH'd to computer B and then from B to computer C, is there a way I would be able to tell short of closing the session and using 'hostname' to discover where I am at?

(Presumably, this solution would also make it clear that I am not SSH'd anywhere, if I am just at localhost but if I am SSH'd elsewhere and from there SSH'd back to my local machine, I'd be able to tell.)

Try setting your prompt to include the hostname; otherwise I like to use screen on every login with a status line containing the hostname (a bit complicated to setup, and it gets recursive with status lines within screens with status lines). You should be able to echo $HOSTNAME or just execute hostname.